Best constants in the Hardy–Rellich inequalities and related improvements

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• 作者：A. Tertikas ; N.B. Zographopoulos
• 关键词：Hardy– ; Rellich inequalities ; Rellich– ; Sobolev inequalities ; Best constants ; Optimal inequalities
• 刊名：Advances in Mathematics
• 出版年：2007
• 出版时间：1 March 2007
• 年：2007
• 卷：209
• 期：2
• 页码：407-459
• 全文大小：382 K

文摘

We consider Hardy–Rellich inequalities and discuss their possible improvement. The procedure is based on decomposition into spherical harmonics, where in addition various new inequalities are obtained (e.g. Rellich–Sobolev inequalities). We discuss also the optimality of these inequalities in the sense that we establish (in most cases) that the constants appearing there are the best ones. Next, we investigate the polyharmonic operator (Rellich and higher order Rellich inequalities); the difficulties arising in this case come from the fact that (generally) minimizing sequences are no longer expected to consist of radial functions. Finally, the successively use of the Rellich inequalities lead to various new higher order Rellich inequalities.